Bimodality in single even-aged population: A simulation study

IN SINGLE

BIMOOALITY

Jan

EVEN-AGED

POPUPTION:

A SIMULATION

STUDY

Lepsl)

The simulation model of the self-thinning even-aged population Summary. was constructed and calibrated on the basis of field data. The aim of the simulation study was to find and biologically interpret the conditions leading to bimodality in distribution of plant heights. These a re : etriklngly asymmetric competition influencing the growth of individuals and high initial density. The results of simulations were compared with generalized 3/2 power law. Key words: thinning, 1.

Int

The

single

systems.

and

phenomena

even-aged They

seem

that

may

even-aged

populations

-thinning

1s

based

on

of

ruderal

2.

Some

(Yoda

field

al.

w is the

the

in

very of

those

often

more

the

self-

some

(usually Our

from

simplest

studying

complicated

studied.

(1982)

of

for

organisms

Prach

ecolog

systems.

In

plants),

self-

simulation

study

self-thinning

it

processes pure

is

populations

populations

i.e.

the may

usually

plant

described

w = k

-c-3/2

weight

of

an

that

follow

the

weight-density

be

interesting

course in

of

of

study

feature

weights.

their

others

by

the

individual Hozumi

3/2

power

law

trajectory

in

the

formula

andris

(1977)

the

seems

to

density. be

of

great

to

the

appears

self-thinning

(e.g. 1s

1s

It

blmodality ln

(e.g.

Mohler,

Marks

elucidate

and

some

some

in

Ford

1975,

Sprugel

possible

distribution

even-aged Prach

1978).

causes

of

populations

of

1982).

it

The

aim

of

the

appearan-

blmodality.

models

Munro

(1974)

growth

models.

growth.

The

distinguished

They

of

the

philosophy

first

and

1) Department 370

05

may

resolution

purposes

delling

702,

obscure sedentary

populatlone

and/or

present

The

suitable

natural

average

other

not

3.

of

1963),

heights

of

the

bimodality,

significance.

The

ce

are

be very

generalizations,

ecological

the

data

self-thinning

Among

does

be of

self-thlnnlng

of

plant

populations to

process

features

where

For

population,

annuals.

et

course

the

the

The

the

plant

law.

reduction

cal

In

even-aged

simulation, 3/2 power

of Ceske

of

other

with

Biomathematics,

decreases

study, and the

modelling applied

models

used

Budltjovice,

three directly

present was

the

be

a simple

calibrated. third

philosophies

on models from

the

model

Btological

to the

Centre,

forest

the

second

models,

philosophy

Research

of population

first

with

Two other modelling

of

one were

third. mowith exani-

Na sadkach

CzechFslovakia

205

ned

for

only

their

the

the

other

1)

The

tion 2)

length

of

The

the 4)

its

simulated

height

unit

The

However,

described

assumptions

canopy

of

is

described

of

other

individual

plot

are

occupied

to

plots

the

individual

the

the

by

square

within

occupied

the

its

here,

the

by logistic

equa-

individuals. an

a particular its

plot

about

neighbours:

considered

of

the by all

comparison

smaller

of

and

allometric

individual

is

height,

is

a

function

individuals

-

a particular

on

with

asymmetric

to

of

the

the.

area

of

the

the

the

individual

status

of

average

taller

the

height

depends

individual of

individuals

the

are

on (i.e.

canopy).

The

influenced

less

ones.

probability

depends

growth

competition

in is

the The

is

competition

competition

of

of

competition

5)

bimodality.

plot.

reduction

strength

than

each

proportional

sum of

feasible

within

of

of

the

of

philosophy

ecologically

for

height. be

strength

on its

of

appearing

mentioned. on

term

the modelling

individual

added

to

of

The

the

based

simulate second

characteristics

function

ratio

is

growth

an

considered 3)

briefly

a single

with Other

the

are

model

of

to

with

two

The growth

ability

model

of

on

the

survival

of

competition

a particular

strength

individual

and

on

the

for

status

of

one

time

the

indi-

vidual, The ly.

The

mathematical model

Their

plot. growth

of Li

I

the

t

rated

as

the

is

length

its

The

term

REDi where

I

t

Lmeant

REDi

= fl is

function,

second

of

one.

survival

to

I

t

the

the

next

The the

second.

ferent

with

Pi , t 6 different

sensitivities

(0,l)

the

point, Lit

simulated

length

t,

ri

ri

and

N (e,

=0.05

is

its

Lmaxi , a;‘)

variability. x@),

for of

growth

were

both

In

The

ruderal

gene-

and

difference annuals

competition

on

1s the

one growth

(3)

) of

the

canopy.

first

variable

forms

were

Pi

is

t,

The and

examined.

fl

is

a non-nega-

increasing

in

the

The

probability

and

increases

(or

the

of

expressed

*

----I

aLg%ytin

the

However,

growth

time

natural

influence

height

parameters) of

6

Li,t Lmeant

functional time

the

the

The

brief-

t

60th

time-step the

in

I

distributions

(usually suitable

average

( L? ‘i t = f* funltion f 2 decrJe>\e(s

tions

at

express

in

very

described

individual

normal

only

growing

is

length.

given

considered.

i-th

L2 J,t’

Oifferent

not

REOi

to

decreasing

is

(l-

describes

(

is

Li,t Lmaxi )*

with

The

model

n individuals

plot

possible

chosen used.

the

(l=l,...,n)

maximal

pd<@were were

of

the

ri.Li,t.

random numbers z si; ) respectively

equations

tive

+

I

of

a set

within individual

LX t is

Li

LmAxi

day.

with

i-th

t+l=

rate,

cases:

deals location

where

N (p,,

formulation

first

variable

different are

and

used survival

functions to

enable

characterizing

towards

competition

(4) in

same

func-

difstress.

The

Monte

ves

to

with

Carlo

the

normal

the

is

data model

based

on

techniques tal

structure.

the

plot.

ting

changes

and

Kindlmann Leslie

divided

into

model height

and

is

into

in

numbers

As

a model

with

of

population

the the

to

both

step

course

of

in

used.

model

to

fit

within

and

survival)

for

simula-

self-thinning

(LepS

philosophy,

The

survival

order

horizon-

point

modelling

was

modelling the

growth

this

and

comparison

account

used

third

growth

Transition

for

Similar

a particular

We have

during

each

one.

(influencing

only.

classes.

taking

located

strength

in

survi-

random

corresponding

used

described

in

pattern

computed

individual as

deviation

philosophy

the

differs

neighbours

1984).

adapted

as

individual

spatial

the

generated

standard

modelling

It

competition

in

whether

were

(1982).

first

employed.

joining

decide

the

assumptions

Each

The

variable

Prach the

to

lengths with

of

similar

on

used

Initial

with

were

depends

was

time.

distribution,

original The

method

next

the

population

was

probabilites the

above

were listed

assumptions.

4.

Results

of

The (1982)

given

show

values king

comparison

is

riments

simulations

and

in

that

the

on

initial

dependence

of

cient

initial

be very

of

These

97.

used weight

of

The

simulation

of

conditions. value

on

The

mortality hold

original

analysis

bimodality

The

REDi

conclusions

with

sensitivity

appearance

the

Fi?flcH( LB%?,

Fig.1

1.

density.

high.

a$

results

Fig.

depends

necessary the of

and

on

conditions

status

of

also

the are

individual

undercanopy

roughly

data

of

expe-

parameter the and

individuals for

Prach

simulation

the

two

strisuffi-

must

not

models

BIllLRTION A wI

ml0

JS

\

1

Results of simulations of the Atriplex nitens population model. Left: Comparison with original data of Prach (1982) - distribution of plant heights at dates 28.3., 3.6.,15;7. Densities are in the upper right corner of each histogram. Right: Simulated weight - density trajectories for different initial densities. The broken line corresponds to -3/2 slope. for

comparison. is

considered

The to

mean be

plant

proportional

weight to

-

density

the

3-rd

trajectory power

of

(the the

length)

207

corresponds law

well

to

Note

(Fig,l).

overcrowded

Discussion

to

explain

There

several bimodality

Aikman

and

on

data

the

that

the

asymmetric

conclusion

let-,

Marks

all

nature1

in

seed

Prach

Ford

models

law

for

with

some

in

the

the

is

the

3/2

intended

power for

may

intensity or

that

consider

to

two

or

in

bimodality

of

(Ford

be

at

1975,

(e.g.

early

is appear. Moh-

obserat

very

appearance be

used

Rabinowicz

cause

the

even-aged.

waves

not

However, the

cannot

(cf. may

are

bimodality

densities

and

them

here

does

initial

asymmetry

even

of

techniques

experience

slowed

several

All

bimodality

bimodality

reasons

1976,

presented

bimodality. be

other we

Gates

1981).

those

high

of

in

trying

(Diggls

appearence

field

distinct

simulational)

modelling

the

strength

which

Diggle

growths

growth

and

including

Extremely

The

results

and

Although

1978).

the

germination

obviously (cf.

of

populations

agreement

resulted

populations,

is

power

analytical

condition

Sprugel

are

(both

In sparse

competition

there

3/2

even-aged

(1975). the

necessary

densities

of

However,

the

1980,

Ford

postponed.

measure

ment

of

(1982)

initial

bimodality

which

Watkinson

is

and

by Prach

high

in

competition.

This

one

models

considerably,

agree

generalization

only.

the

1978,

s (1877)

originally

are

based

ved

that

Horumi

populations

5.

differ

the

of as

1979).

bimodality The

due

stages

most

to

of

in commor

weather)

develop-

1982).

Literature Aikman, D.P., and A.R. Watkinson: A model for growth and self-thinning in even-aged monocultures of plants. Ann. Bot. 45 (1980), 419-427. stochastic model of inter-plant competition, Diggle,.P.3. : A spatial J. Appl. Prob. 13 (1976), 662-671. Ford, E.D.: Competition and stand structure in some even-aged plant monocultures. J. Ecol. 63 (19751. 311-333. Ford, E.D., and P.J. Diggle: Competition for light in a plant monoculture modelled as a spatial stochastic process. Ann. Bot. 48 481- 500. (1981), Bimodality in even-aged plant monocultures. J. Theor. Gates, D.J.: Biol. 71 ( 1978), 525-540. K.: Ecological and mathematical considerations on self-thin+ Hozumi, nina in even-saed oure stands. II. Growth analysis of self-thinnina. _. Bat-Msg. Tokyo”93 (1980), 149-166. Mohler, C.L., P.L. Marks, and D.G. Sprugel: Stand structure and allometry of trees during self-thinning of pure stands. 2. Ecol. 66 (1978). 599-614. Forest growth models - a prognosis. In: J. Fries ed. : Munro,D.: Growth models for tree and stand simulation. Royal College of Forestry, Stockholm, pp. 7 - 21, 1974. in selected ruder-al species K. : Self-thinning processes Prsch, populations. In Czech. Preslia 54 (1982)) 271-275. in distributions of seedling weight Rabinowicz, 0. : Bimodal relation to density of Festuca paradoxa Desv. Nature 277 (1979)) 297-298,

Yoda, K., T. Kira, H. Ogawa, and K. Hozumi: Self-thinning in overcrowded pure stands under cultivated and natural conditions. J. Biol. Osaka City Univ. 14 (1963). 107-129.